Recall the binomial theorem:
[tex]\displaystyle (a + b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k[/tex]
where [tex]\binom nk = \frac{n!}{k!(n-k)!}[/tex] is the binomial coefficient.
Take a = x/3, b = -3, and n = 7. Then we get the x⁵ and x⁴ terms when 7 - k = 5 and 7 - k = 4, respectively; or when k = 2 and k = 3.
[tex]k=2 \implies \dbinom 72 \left(\dfrac x3\right)^{7-2} (-3)^2 = 21 \cdot \dfrac{x^5}{243} \cdot 9 = \dfrac79 x^5[/tex]
[tex]k=3 \implies \dbinom 73 \left(\dfrac x3\right)^{7-3} (-3)^3 = 35 \cdot \dfrac{x^4}{81} \cdot (-27) = -\dfrac{35}3 x^4[/tex]
Then when multiplying this expansion by x - 6, we get an x⁵ terms from the products
[tex]\dfrac79 x^5 \cdot (-6)[/tex]
and
[tex]-\dfrac{35}3 x^4 \cdot x[/tex]
so that the x⁵ term in the overall expansion of (x/3 - 3)⁷ (x - 6) has a coefficient of
[tex]\dfrac79\cdot(-6) + \left(-\dfrac{35}3\right) \cdot 1 = \boxed{-\dfrac{49}3}[/tex]
A company has two large computers. The slower computer can send all the company’s email in 50 minutes. The faster computer can complete the same job in 30 minutes. If both computers are working together, how long will it take them to do the job?
Answer:
18 minutes and 45 seconds
Step-by-step explanation:
[tex]\frac{1}{50}[/tex] + [tex]\frac{1}{30}[/tex] = [tex]\frac{3+5}{150}[/tex] = [tex]\frac{4}{75}[/tex]
1 ÷ [tex]\frac{4}{75}[/tex] = [tex]\frac{75}{4}[/tex] = 18' 45''
Answer:
18.75 minutes If A and B work together
a prisms base has a perimeter of 12cm and a height of 2 cm. the area of the base was 5cm. what is the surface area of the prism.
==============================================================
Explanation:
For any prism, there are two congruent bases opposite one another. In other words, the bases are identical polygons with the same area. Each base is 5 square cm. So far, that accounts for 5+5 = 10 square cm of surface area. Let A = 10 as we'll use it later.
The perimeter is 12 cm and the height is 2 cm. It turns out that the lateral surface area is found by multiplying the two items. This is because the perimeter is made up of the lengths of the sides of the base, which when multiplied by the height, will give the area of each rectangular side panel.
So the lateral surface area is 12*2 = 24 sq cm. Let B = 24.
Overall, the total surface area is A+B = 10+24 = 34 square cm
One kilometre is about 0.62 miles. It is 12 kilometres from Noah's house to the
ice skating rink. About how many miles is it from Noah's house to the ice
skating rink?
u'll get extra points if u answer guys!! pls help!!!!!
Answer:
7.44miles
Step-by-step explanation:
1km = 0.62miles
12km will be ; 0.62 × 12
= 7.44miles
Billy invests $5,000 in an account at 4.2% for 15 years, compounded monthly. Assuming no other deposits or withdrawals are made, how much will Billy have at the end of 15 years?
The amount Billy has at the end of the 15 years is $9,377.73.
What is compound interest?It is the interest we earned on the interest.
The formula for the amount earned with compound interest after n years is given as:
A = P[tex](1 + r/n)^{nt}[/tex]
P = principal
r = rate
t = time in years
n = number of times compounded in a year.
We have,
P = $5000
r = 4.2%
t = 15 years
n = 12
A = P[tex](1 + r/n)^{nt}[/tex]
A = 5000 [tex](1 + 0.0035)^{180}[/tex]
A = 5000 x [tex]1.0035^{180}[/tex]
A = $9,377.73
Thus,
Billy has $9,377.73 at the end of the 15 years.
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-3 (-3 - n) - 4 = 3n + 5
[tex]-3 (-3 - n) - 4 = 3n + 5\\\\\implies 9+3n -4 = 3n+5\\\\\implies 5 +3n = 3n+5\\\\\text{The equation is true for all values of n.}[/tex]
The length of my rectangular calculator lid is 11 mm more than the width. The area is 152 square
mm. Find the dimensions of the lid.
Answer:
The dimensions of the lid are 8mm by 19mm.
w = 8
l = 19
Step-by-step explanation:
[tex]l=w+11\\l\times w=152[/tex]
where l is length and w is width. This can be solved as a system of equations.
[tex]l\times w=152\\(w+11)\times w=152\\w(w+11)=152\\w^2+11w=152\\w^2+11w-152=0[/tex]
At this point, it gets a little tough. I might be unnecessarily overcomplicating things, but this is the only way I see to solve the problem.
=================== Skip down below if you don't care about factoring
You need to factor the newly created trinomial.
[tex]ax^2+bx+c[/tex]
With a trinomial in this form, you need to find 2 numbers that add together to make b and multiply together to make ac.
Here, we need 2 numbers that add to 11 and multiply to -152. First, factor 152:
1, 152
2, 76
4, 38
8, 19
Then the reverse of all of those is true too, of course:
19, 8
38, 4
etc
In our case, we're looking for -152, so one of our factors will be negative. We're also looking for factors that add up to 11. Looking at these factors, you can see that 19 - 8 = 11, so our factors are 19 and -8.
Finally, you can use those to factor our trinomial. Split up the middle number (11w) into two:
[tex]w^2+11w-152=0\\w^2-8w+19w-152=0[/tex]
And now, you can factor by grouping:
[tex]w^2-8w+19w-152=0\\w(w-8)+19(w-8)=0\\(w+19)(w-8)=0[/tex]
===================
Now that the number is factored, you can finally find w:
[tex](w+19)(w-8)=0[/tex]
Here, you can see that the equation will be true when w = -19 or w = 8. Those are our solutions, but we can't have a negative distance, so it's just
[tex]w=8[/tex]
Going all the way back to the top, now you can use the width to find the length.
[tex]l=w+11\\l=8+11\\l=19[/tex]
That one was much easier.
The dimensions of the lid are 8mm by 19mm.
Finally, check that with both of the original equations to make sure it's correct.
[tex]l=w+11\\19=8+11\\19=19\\\\l\times w=152\\19\times8=152\\152=152[/tex]
Find the measure of angle m
Given a conditional statement p → q, find the inverse of its inverse, the inverse of its converse, and the inverse of its contrapositive.
The inverse of an implication p ⇒ q is ¬p ⇒ ¬q.
The converse of p ⇒ q is q ⇒ p.
The contrapositive of p ⇒ q is ¬q ⇒ ¬p.
Then
• the inverse of the inverse of p ⇒ q is p ⇒ q
• the inverse of the the converse of p ⇒ q is ¬q ⇒ ¬p
• the inverse of the contrapositive of p ⇒ q is q ⇒ p
if EF = 8x + 16, FG = 60, and eg = 156, find the value of x
Answer:
10
Step-by-step explanation:
8x+16+60=156
8x+76=156
8x=80
x=10
Find the value of x.
Answer:
9
Step-by-step explanation:
yeah, 9...........right???
A computer can do 1000 operations in 4.5×10^-6 seconds. How many operations can be done by this computer in one hour?
Answer:
0.00027.
Step-by-step explanation:
All you have to do is do the scientific notation then multiply by 60.
Which best describes reduced fractions that will always terminate?
A.
reduced fractions with denominators that are factors of only 5
B.
reduced fractions with denominators that are factors of 2 or 5
C.
reduced fractions with denominators that are factors of only 5 or 7
D.
reduced fractions with denominators that are factors of only 2 or 3
Answer:a
Step-by-step explanation:
Reduced fractions with denominators that are factors of 2 or 5. Therefore, the correct answer is option B.
Terminating decimals are integers with a definite or finite number of digits following the decimal point. Like fractions, decimal numbers are used to represent a portion of a whole. This course will concentrate on the type of decimal numbers known as terminating decimal numbers. The word 'terminate' means to put a stop to something. In decimal terms, it is a number that comes to an end.
Here are a few tips to assist you identify a terminating decimal number.
A non-rational number is never an ending decimal number.If the denominator of a simplified rational number can be expressed in the manner [tex]2^p5^q[/tex] or [tex]2^p[/tex] or [tex]5^q[/tex], where p, q N, then the number has a terminating decimal expansion.After the decimal point, a terminating decimal number always has a finite number of digits.Therefore, the correct answer is option B.
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13. Solve to find the vertex if :
3x2 – 19x - 5
Answer:
A) y = 3(x -3)^2 -46
B) (3, -46)
C) look at the y-coordinate of the vertex
Step-by-step explanation:
A) Factor the leading coefficient from the variable terms.
y = 3(x^2 -6x) -19
Inside parentheses, add the square of half the x-coefficient. Outside, subtract the same value.
y = 3(x^2 -6x +9) -19 -3(9)
y = 3(x -3)^2 -46
__
B) Compared to the vertex form, ...
y = a(x -h)^2 +k
we find a=3, (h, k) = (3, -46).
The vertex is (3, -46).
__
C) The vertex is an extreme value (as is any vertex). The sign of the leading coefficient tells you whether the parabola opens upward (+) or downward (-). This parabola opens upward, so the vertex is a minimum.
If the leading coefficient is positive, the y-coordinate of the vertex is a minimum. If the leading coefficient is negative, the y-coordinate of the vertex is a maximum.
Step-by-step explanation:
how do i do math in division fractions?
Answer:
The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. Finally, simplify the fractions if needed.
Answer:
Keep change flip!
Step-by-step explanation:
Example 1/2 divided 3/4
We simply keep 1/2 and then we flip the 3/4 which is now 4/3
now we also flip the division sign which its opposite is multiplying now its simply 1/2 x 4/3 and you get 4/6. Hope this helps please mark brainliest!
If ∆ ≅ ∆, = 5 − 4 and = 8 − 10, find XZ.
Please show work!
Answer:
6
Step-by-step explanation:
Two triangles are congruent so we can use the similarity ratio to find the length of XZ
8x - 10 = 5x - 4 export like terms to the same side of equation
8x - 5x = 10 - 4
3x = 6 divide both sides by 3
x = 2 and so the length of XZ is 8*2 - 10 = 6
What is the greatest common factor shared by 40 and 20?
Answer here
SUBMIT
Answer:
20
Step-by-step explanation:
factors of 20 = 1 2 4 5 10 20
factors of 40 = 1 2 4 5 8 20 10
therefore 20 is the greatest common
Helppppp pleaseee????
Answer:
Lower quartile 3, median 5, Upper quartile 7
Step-by-step explanation:
First divide the data in half 1 3 4 and 6 7 7
The median is the middle between 4 and 6, so it is 5.
The lower quartile is the median in the lower half, 3
The upper quartile is the median in the upper half, 7
If I have a 1/4 chance of winning a bet, but lose it 40 times in a row, what percentage chance is that?
The average annual cost (including tuition, room, board, books, and fees) to attend a public college takes nearly a third of the annual income of a typical family with college-age children (Money, April 2012). At private colleges, the average annual cost is equal to about 60% of the typical family's income. The following random samples show the annual cost of attending private and public colleges.
Private colleges: 52.8, 43.2, 45.0, 33.3, 44.0, 30.6, 45.8, 37.8, 50.5, 42.0.
Public colleges: 20.3, 22.0, 28.2, 15.6, 24.1, 28.5, 22.8, 25.8, 18.5, 25.6, 14.4, 21.8.
Required:
a. Based on your computation of the two sample means and the two sample standard deviations, find the degrees of freedom.
b. What is the point estimate of the difference between the two population means?
c. Develop a 95% confidence interval of the difference between the annual cost of attending private and pubic colleges.
Schedule of Cash Payments
Shakti Power Inc. was organized on February 28. Projected selling and administrative expenses for each of the first three months of operations are as follows:
March
$68,000
April
77,000
May
88,000
Depreciation, insurance, and property taxes represent $5,000 of the estimated monthly expenses. The annual insurance premium was paid on February 28, and property taxes for the year will be paid in Online file. Open the spreadsheet, perform the required analysis, and input your answers in the question below.m
dAnswer:
fffff
Step-by-step explanation:ff m
PLEASE HELP ME THIS IS THE 5TH TIME I HAVE HAD TO POST THIS AND GIVING BRAINLIST TO FIRST ANSWER
B.
C.
D.
Answer:
C.
Step-by-step explanation:
- z > 8 equivalent inequality
Answer: z < -8
Step-by-step explanation:
Since z is negative, divide both sides by -1, which leaves you with z > -8.
Multiplying or dividing an inequality by a negative number flips the sign, thus the answer is z < 8.
Correct me if I am incorrect.
The factor that most heavily influences your credit score is
A. length of credit history
B. types of credit
C. amount owed
D. payment history
Answer:
D.payment history
Step-by-step explanation:
Help help math math math
Answer:
-3/7,0
Step-by-step explanation:
This morning, Raina's car had 23.21 gallons of fuel. Now, 1.5 gallons are left. How much fuel did Raina use?
Help help help math math math
Answer:
yes this right cuz ye or nah?
Step-by-step explanation:
yes cuz 0 1 2 3 4 5 6 7 8 9 are the numbers
Answer: [?] is 6
Step-by-step explanation:
x=5
so, 2x=10
10-4 is 6
130 is what percent of 70?
Answer:
91%
Step-by-step explanation:
hopes this helps
Answer:
185.71%
Step-by-step explanation:
130 / 70 = 1.857142857
(1.857142857)(100) = 185.71%
1. Divide
(7x2 - 13x+4) - (x-1
)
Answer:7 x 2 − 14 x + 5
Step-by-step explanation:
1 In general, given a{x}^{2}+bx+cax
2
+bx+c, the factored form is:
a(x-\frac{-b+\sqrt{{b}^{2}-4ac}}{2a})(x-\frac{-b-\sqrt{{b}^{2}-4ac}}{2a})
a(x−
2a
−b+
b
2
−4ac
)(x−
2a
−b−
b
2
−4ac
)
2 In this case, a=7a=7, b=-14b=−14 and c=5c=5.
7(x-\frac{14+\sqrt{{(-14)}^{2}-4\times 7\times 5}}{2\times 7})(x-\frac{14-\sqrt{{(-14)}^{2}-4\times 7\times 5}}{2\times 7})
7(x−
2×7
14+
(−14)
2
−4×7×5
)(x−
2×7
14−
(−14)
2
−4×7×5
)
3 Simplify.
7(x-\frac{14+2\sqrt{14}}{14})(x-\frac{14-2\sqrt{14}}{14})
7(x−
14
14+2
14
)(x−
14
14−2
14
)
4 Factor out the common term 22.
7(x-\frac{2(7+\sqrt{14})}{14})(x-\frac{14-2\sqrt{14}}{14})
7(x−
14
2(7+
14
)
)(x−
14
14−2
14
)
5 Simplify \frac{2(7+\sqrt{14})}{14}
14
2(7+
14
)
to \frac{7+\sqrt{14}}{7}
7
7+
14
.
7(x-\frac{7+\sqrt{14}}{7})(x-\frac{14-2\sqrt{14}}{14})
7(x−
7
7+
14
)(x−
14
14−2
14
)
6 Simplify \frac{7+\sqrt{14}}{7}
7
7+
14
to 1+\frac{\sqrt{14}}{7}1+
7
14
.
7(x-(1+\frac{\sqrt{14}}{7}))(x-\frac{14-2\sqrt{14}}{14})
7(x−(1+
7
14
))(x−
14
14−2
14
)
7 Remove parentheses.
7(x-1-\frac{\sqrt{14}}{7})(x-\frac{14-2\sqrt{14}}{14})
7(x−1−
7
14
)(x−
14
14−2
14
)
8 Factor out the common term 22.
7(x-1-\frac{\sqrt{14}}{7})(x-\frac{2(7-\sqrt{14})}{14})
7(x−1−
7
14
)(x−
14
2(7−
14
)
)
9 Simplify \frac{2(7-\sqrt{14})}{14}
14
2(7−
14
)
to \frac{7-\sqrt{14}}{7}
7
7−
14
.
7(x-1-\frac{\sqrt{14}}{7})(x-\frac{7-\sqrt{14}}{7})
7(x−1−
7
14
)(x−
7
7−
14
)
10 Simplify \frac{7-\sqrt{14}}{7}
7 7− 14 to 1-\frac{\sqrt{14}}{7}1− 7 14
7(x-1-\frac{\sqrt{14}}{7})(x-(1-\frac{\sqrt{14}}{7}))
7(x−1− 7 /14 )(x−(1− 7 14 )) 11 Remove parentheses.
7(x-1-\frac{\sqrt{14}}{7})(x-1+\frac{\sqrt{14}}{7})
7(x−1− 7/ 14 .7(x−1+ 7 14 )
During a sale, a store offered a 15% discount on a camera that originally sold for $690. After the sale, the discounted price of the camera was marked up by 15%. What was the price of the camera after the markup? Round to the nearest cent.
Answer:674.475
Step-by-step explanation:
(690)(0.15)= 103.5
690-103.5= 586.5 (after first markdown)
(586.5)(0.15)= 87.975
586.5+87.975 (Final price after markup)
solve the log equation
Answer:
x = -5
Step-by-step explanation:
log5((5-3x)/((7+x)^2))=1
(5-3x)/(7+x)^2 = 5
5-3x = 5x^2+70x+245
5x^2+73x+240=0
x= -5, x= -48/5
-48/5 is extraneous
x = -5